Linear pulsations and stability of differentially rotating stellar models. II. General relativistic analysis

The author has previously given an Eulerian velocity-potential variational principle for relativistic perfect-fluid hydrodynamics. The second variation of the principle is here used as the Lagrangian density governing the evolution of small perturbations of fully relativistic, differentially rotating stellar models. Noether's theorem is used to construct a globally conserved angular-momentum density, whose integral over a spacelike hypersurface is the second-order correction to the star's total angular momentum. From the Harniltonian is constructed a globally conserved energy density, whose integral is the correction to the star's active gravitational mass. By Liapunov's second theorem, positive-definiteness of the energy density guarantees stability of the star. In the Newtonian limit and in the special case of relativistic radial pulsations, this is equivalent to stability criteria already Imown. Means are discussed whereby the general criterion might be made more suitable for practical applications.